Stochastic Evolutionary Stability in Extensive
Form Games of Perfect Information.
Abstract: Nöldeke and Samuelson
(1993) investigate a stochastic evolutionary model for extensive form games and
show
that even for games of perfect information with a unique subgame
perfect equilibrium, non-subgame perfect equilibrium-strategies
may well survive in the long run even when mutation rates tend to
zero. In a different model of evolution in the agent normal form
of these games Hart (2002) shows that under suitable limit-taking,
where small mutation rates are accompanied by large population
sizes in a particular way, the unique prediction is again the
subgame perfect equilibrium. This paper provides a proof of a
similar result for the model of Nöldeke and Samuelson (1993).
JEL classification: C62, C72, C73; Keywords: backward induction, learning, experimentation,
subgame-perfect equilibrium
Tail-dependence in stock-return pairs (with Ines Fortin).
Abstract: The empirical joint distribution of
return-pairs on stock indices displays high tail dependence in the lower tail.
The presence of tail dependence is not compatible with the assumption of (conditional)
joint normality. A general test for one dependence structure versus another via
the profile likelihood is described and employed in a bivariate GARCH model,
where the joint distribution of the disturbances is split into its marginals
and its copula. The copula used in the paper is such that it allows for the possibility
of the existence of lower tail-dependence and such that it encompasses the normal
copula. The model is estimated using bivariate data on a set of European stock
indices. The assumption of 'normal' dependence is easily rejected in favour of
a more tail dependent distribution.
JEL classification: C12, C32, C51, C52, G15; Keywords: Value-at-Risk,
copula, non-normal bivariate GARCH, asymmetric dependence, profile likelihood-ratio
test
Optimal window-width choice in spectral
density estimation: Review and simulation (with
Ines Fortin).
Abstract: This paper deals with optimal window width
choice in non-parametric lag or spectral window estimation of the spectral density
of a stationary zero-mean process. Several approaches are reviewed: cross-validation-based
methods as described by Hurvich (1985), Beltrão& Bloomfield
(1987) and Hurvich & Beltrão (1990); an iterative procedure developed
by Bühlmann (1996); and a bootstrap approach followed by Franke & Härdle
(1992). These methods are compared in terms of the mean square error, the mean
square
percentage error, and a third measure of the distance between the true spectral
density and its estimate. The comparison is based on a simulation study, the
simulated processes being in the class of ARMA(5,5) processes. On the basis of
simulation evidence we suggest to use a slightly modified version of Bühlmann's
(1996) iterative method. This paper also makes a minor correction of the bootstrap
criterion by Franke & Härdle (1992).
Efficient Risk-Sharing Rules with Heterogeneous
Risk Attitudes and Background Risks (with Chiaki Hara). [Pdf]
Abstract: In
an exchange economy in which there is a complete set of markets for macroeconomic
risks but no market for idiosyncratic risks, we consider how the efficient
sharing rules for the macroeconomic risk are affected by the heterogeneity
in the consumers' risk attitudes and idiosyncratic risks. While the curvature
of the risk-sharing rules at high consumption levels are governed by the consumers'
risk attitudes, the curvature at low consumption levels depend not only on
the risk attitudes but also on the lower tail distributions of the idiosyncratic
risks.
JEL classification: D51, D58, D81, G11, G12, G13; Keywords:
Efficient risk-sharing rules, absolute risk tolerance, idiosyncratic risks,
background risks, Inada condition
Representative consumer's rsk aversion and efficient
risk-sharing rules (with Chiaki Hara). [Pdf]
Abstract: We
study the representative consumer's risk attitude and efficient risk-sharing
rules in a single-period, single-good economy in which consumers have homogeneous
probabilistic beliefs but heterogeneous risk attitudes. We prove that if all
consumers have convex absolute risk tolerance, so must the
representative consumer. We also identify a relationship between the curvature
of an individual consumer's individual risk sharing rule and his absolute cautiousness,
the first derivative of absolute risk-tolerance. Some consequences of these
results and refinements of these results for the class of HARA utility functions
are discussed.
JEL classification: D51, D58, D81, G11, G12, G13; Keywords:
aggregation, heterogeneous consumers, absolute risk tolerance, mutual fund
theorem
On the elimination of dominated strategies in stochastic
models of evolution with large populations. [Pdf]
Abstract: This paper analyzes a stochastic
best reply evolutionary model with inertia in normal form games. The long-run
behavior of individuals in this model is investigated in the limit where experimentation
rates tend to zero, while the expected number of experimenters, and hence
also population sizes, tend to infinity. Conditions on the learning-rate which
are necessary and sufficient for the evolutionary elimination of weakly dominated
strategies are found. The key determinant is found to be the sensitivity of the learning-rate to small payoff differences.
JEL classification: C62, C72, C73; Keywords: learning, experimentation,
$S^{\infty}W$-procedure, weak dominance, iterated strict dominance
Individual and group selection in symmetric 2-player games.
[Pdf]
Abstract: A population of individuals is divided into groups. Individuals are recurrently randomly matched with individuals from their group to play a generic symmetric 2-player game. Deterministic inter- and intra-group dynamics are derived from a model of individual imitation within groups and individual migration between groups. Conditions are identified under which subsets (components) of the set of stationary states are (interior) asymptotically stable. The results are then applied to generic coordination games and the prisoners' dilemma. The unique asymptotically stable set of stationary states in coordination games is such that every individual in non-extinct groups plays the Pareto-optimal equilibrium. In the multi-group prisoners' dilemma there is no sensible asymptotically stable set of states.
JEL classification: C62, C72, C73; Keywords: equilibrium selection,
payoff monotonic
dynamics, inter- and intra-group dynamics, local interaction,
hierarchical evolution, multilevel evolution, imitation, experimentation
Refined best-response correspondence
and dynamics (with Dieter Balkenborg and Josef Hofbauer).
Abstract:
We study a natural (and, in a well-defined sense, minimal) refinement
of the best-reply correspondence. Its fixed points, notions of rationalizability
and CURB sets based on this correspondence are defined and characterized. Finally
minimally asymptotically stable faces under the induced refined best-reply
dynamics are characterized.
JEL classification: C62, C72, C73; Keywords: Evolutionary game theory, best
response dynamics, CURB sets, persistent retracts, asymptotic stability, Nash equilibrium refinements, learning